Finance Formulas - Condensed Cheat Sheet

1. Annuities and Cash Flows

🔑 Key Words: "monthly payments", "forever", "starting next year", "effective annual interest", "starting in year X"

Present Value of Ordinary Annuity: PV = PMT × [(1 - (1 + r)^(-n)) / r]

Future Value of Annuity: FV = PMT × [((1 + r)^n - 1) / r]

Delayed Annuity (starting in year k): PV = PMT × PVIFA(r,n) / (1+r)^k

Perpetuity: PV = PMT / r

Growing Perpetuity: PV = CF₁ / (r - g) (when r > g)

Interest Rate Conversions:

• Annual to monthly: r_monthly = (1 + r_annual)^(1/12) - 1
• Effective interest rate: r_eff = (1 + r_nom/m)^m - 1

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2. NPV and IRR

🔑 Key Words: "independent of macro variables", "two IRRs", "should be taken if"

Basic NPV: NPV = Σ [CFt / (1+r)^t] - C₀

Special IRRs:

• Perpetual project: IRR = CF_annual / Investment
• Growing perpetuity: CF₁ / (IRR - g) = Investment

Projects with Multiple IRRs:

• If CF₀ > 0 (positive cash flow today): Accept if cost of capital is between the two IRRs
• If CF₀ < 0 (investment): Accept if cost of capital is outside the range of the two IRRs

NPV with CAPM: Discount rate = Rf + β(Rm - Rf)

• "Independent of macro variables" → β = 0 → r = Rf

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3. Bonds

🔑 Key Words: "face value", "coupon", "yield to maturity", "time to maturity"

Bond Price: P = Σ [Coupon/(1+YTM)^t] + [Face Value/(1+YTM)^n]

Perpetual Bond: P = Coupon / YTM

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4. CAPM and SML

🔑 Key Words: "beta", "Jensen's alpha", "CAPM holds", "above/below SML", "correlation with market"

CAPM Formula: E(Ri) = Rf + βi[E(Rm) - Rf]

Beta: βi = Cov(Ri, Rm) / Var(Rm) = ρim × (σi/σm)

Portfolio Beta: βp = Σ wi × βi

Jensen's Alpha: α = E(R) - [Rf + β(E(RM) - Rf)]

• α > 0: Stock above SML
• α < 0: Stock below SML

Calculations from Scenario Data:

• E(R) = Σ [Pi × Ri]
• Var(R) = Σ [Pi × (Ri - E(R))²]
• Cov(RA, RM) = Σ [Pi × (RAi - E(RA)) × (RMi - E(RM))]

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5. Statistics and Portfolio Theory

🔑 Key Words: "independent", "correlation", "standard deviation", "proportion"

Expected Return and Variance:

• E(R) = Σ [Pi × Ri]
• Var(R) = Σ [Pi × (Ri - E(R))²]
• σ = √Var(R)

Covariance and Correlation:

• Cov(X,Y) = Σ [Pi × (Xi - E(X)) × (Yi - E(Y))]
• ρ(X,Y) = Cov(X,Y) / (σX × σY)

Portfolio Formulas:

• E(Rp) = Σ wi × E(Ri)
• σp = √[w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂] (for 2 assets)

Special Cases:

• Independent assets (ρ = 0): σp = √[Σ wi² × σi²]
• Equal weights and independent: σp = σi / √n

Tangency Portfolio: wi = [E(Ri) - Rf] × σj² - [E(Rj) - Rf] × σij / [denominator sum]

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6. Performance Measures

🔑 Key Words: "diversify with risk-free asset", "stock-picking ability"

Sharpe Ratio: S = [E(Rp) - Rf] / σp

Treynor Ratio: T = [E(Rp) - Rf] / βp

When to Use:

• "Diversify with risk-free asset" → Sharpe
• "Stock-picking ability" / "Superior performance" → Jensen's Alpha

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7. Gordon Growth Model

🔑 Key Words: "dividend", "grow by X%", "cost of capital", "decrease at rate"

Stock Price with Constant Growth: P = D₁ / (r - g)

Cost of Equity: r = (D₁ / P) + g

Special Cases:

• Negative growth: g < 0, but still r > g
• No growth: P = D / r

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8. Additional Formulas

🔑 Key Words: "market capitalization", "shortselling allowed", "call/put option"

Time Value of Money:

• FV = PV × (1 + r)^n
• PV = FV / (1 + r)^n

Portfolio Weights:

• Negative weights = Short position
• Sum of weights > 1 = Leveraged investment

Options:

• Call: Max(S-K, 0)
• Put: Max(K-S, 0)

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